I'm studying the Shor algorithm as part of my thesis and have a question about the "measured" phases after the QPE.
So, I take the controlled-U operations on the second register and in cause of phase kickback the relative phase of the controll qubit in register one will change with a multiple of the eigenvalue of $U$. I understand, that $cU$ has multiple eigenvalues with a factor $s$. How can be guaranteed that each of the controlled-Us will kickback the same eigenvalue? Or, why it is not important?
Second, if I run the controlled-U operations and make the QPE, why it is possible to get different results? I thought that the transformation between the bases is unique. So, if my controlled-U makes a specific "change" on the quibit, how it is possible that the QPE generates a superposition with specific probabilities? (e.g. in Nielsen/Chuang Box 5.4 the final measurement will give 0, 512, 1024, 1536)
Thank you for your help.